Abstract
This paper concerns the existence of nontrivial solutions for a boundary value problem with integral boundary conditions by topological degree theory. Here the nonlinear term is a signchanging continuous function and may be unbounded from below.
1 Introduction
Consider the following SturmLiouville problem with integral boundary conditions
where , , , , , α and β are right continuous on , left continuous at and nondecreasing on with ; , and denote the RiemannStieltjes integral of u with respect to α and β, respectively. Here the nonlinear term is a continuous signchanging function and f may be unbounded from below, with is continuous and is allowed to be singular at .
Problems with integral boundary conditions arise naturally in thermal conduction problems [1], semiconductor problems [2], hydrodynamic problems [3]. Integral BCs (BCs denotes boundary conditions) cover multipoint BCs and nonlocal BCs as special cases and have attracted great attention, see [414] and the references therein. For more information about the general theory of integral equations and their relation with boundary value problems, we refer to the book of Corduneanu [4], Agarwal and O’Regan [5]. Yang [6], Boucherif [8], Chamberlain et al.[10], Feng [11], Jiang et al.[14] focused on the existence of positive solutions for the cases in which the nonlinear term is nonnegative. Although many papers investigated twopoint and multipoint boundary value problems with signchanging nonlinear terms, for example, [1520], results for boundary value problems with integral boundary conditions when the nonlinear term is signchanging are rarely seen except for a few special cases [7,12,13].
Inspired by the above papers, the aim of this paper is to establish the existence of nontrivial solutions to BVP (1.1) under weaker conditions. Our findings presented in this paper have the following new features. Firstly, the nonlinear term f of BVP (1.1) is allowed to be signchanging and unbounded from below. Secondly, the boundary conditions in BVP (1.1) are the RiemannStieltjes integral, which includes multipoint boundary conditions in BVPs as special cases. Finally, the main technique used here is the topological degree theory, the first eigenvalue and its positive eigenfunction corresponding to a linear operator. This paper employs different conditions and different methods to solve the same BVP (1.1) as [7]; meanwhile, this paper generalizes the result in [17] to boundary value problems with integral boundary conditions. What we obtain here is different from [620].
2 Preliminaries and lemmas
Let be a Banach space with the maximum norm for . Define and . Then P is a total cone in E, that is, . denotes the dual cone of P, namely, . Let denote the dual space of E, then by Riesz representation theorem, is given by
We assume that the following condition holds throughout this paper.
(H_{1}) is the uniquesolution of the linear boundary value problem
Let solve the following inhomogeneous boundary value problems, respectively:
Lemma 2.1 ([7])
If (H_{1}) and (H_{2}) hold, then BVP (1.1) is equivalent to
whereis the Green function for (1.1).
Define an operator as follows:
It is easy to show that is a completely continuous nonlinear operator, and if is a fixed point of A, then u is a solution of BVP (1.1) by Lemma 2.1.
For any , define a linear operator as follows:
It is easy to show that is a completely continuous nonlinear operator and holds. By [7], the spectral radius of K is positive. The KreinRutman theorem [21] asserts that there are and corresponding to the first eigenvalue of K such that
and
Here is the dual operator of K given by:
The representation of , the continuity of G and the integrability of h imply that . Let . Then , and (2.4) can be rewritten equivalently as
Lemma 2.2 ([7])
If (H_{1}) holds, then there issuch thatis a subcone ofPand.
Lemma 2.3 ([22])
LetEbe a real Banach space andbe a bounded open set with. Suppose thatis a completely continuous operator. (1) If there iswithsuch thatfor alland, then. (2) Iffor alland, then. Here deg stands for the LeraySchauder topological degree inE.
Lemma 2.4Assume that (H_{1}), (H_{2}) and the following assumptions are satisfied:
(C_{1}) There exist, andsuch that (2.3), (2.4) hold andKmapsPinto.
(C_{2}) There exists a continuous operatorsuch that
(C_{3}) There exist a bounded continuous operatorandsuch thatfor all.
(C_{4}) There existandsuch thatfor all.
Let, then there existssuch that
Proof Choose a constant . From (C_{2}), for , there exists such that implies
Now we shall show
provided that R is sufficiently large.
In fact, if (2.7) is not true, then there exist and satisfying
Since , , . Multiply (2.8) by on both sides and integrate on . Then, by (C_{4}), (2.5), we get
Thus,
By (2.9), holds. Then (2.3), (2.6) and (2.10) imply
(C_{3}) shows and (C_{1}) implies . Then (C_{1}), (2.8) and Lemma 2.2 tell us that
It follows from (2.6), (2.11) and (2.12) that
Since , then (2.13) deduces that (2.7) holds provided that R is sufficiently large such that . By (2.13) and Lemma 2.3, we have
□
3 Main results
Theorem 3.1Assume that (H_{1}), (H_{2}) hold and the following conditions are satisfied:
(A_{1}) There exist two nonnegative functionswithand one continuous even functionsuch thatfor all. Moreover, Bis nondecreasing onand satisfies.
Hereis the first eigenvalue of the operatorKdefined by (2.2).
Then BVP (1.1) has at least one nontrivial solution.
Proof We first show that all the conditions in Lemma 2.4 are satisfied. By Lemma 2.2, condition (C_{1}) of Lemma 2.4 is satisfied. Obviously, is a continuous operator. By (A_{1}), for any , there is such that when , holds. Thus, for with , holds. The fact that B is nondecreasing on yields for any , . Since is an even function, for any and , holds, which implies for . Therefore,
that is, . Take , for any , where . Obviously, holds. Therefore H satisfies condition (C_{2}) in Lemma 2.4.
Take and for , , then it follows from (A_{1}) that
which shows that condition (C_{3}) in Lemma 2.4 holds.
By (A_{3}), there exist and a sufficiently large number such that
Combining (3.1) with (A_{1}), there exists such that
and so
Since K is a positive linear operator, from (3.2) we have
So condition (C_{4}) in Lemma 2.4 is satisfied.
According to Lemma 2.4, we derive that there exists a sufficiently large number such that
From (A_{4}) it follows that there exist and such that
Thus
Next we will prove that
If there exist and such that . Let . Then and by (3.4), . The nth iteration of this inequality shows that (), so , that is, . This yields , which is a contradictory inequality. Hence, (3.5) holds.
It follows from (3.5) and Lemma 2.3 that
By (3.3), (3.6) and the additivity of LeraySchauder degree, we obtain
So A has at least one fixed point on , namely, BVP (1.1) has at least one nontrivial solution. □
Corollary 3.1Using () instead of (A_{1}), the conclusion of Theorem 3.1 remains true.
() There exist three constants, andsuch that
Competing interests
The authors declare that no conflict of interest exists.
Authors’ contributions
All authors participated in drafting, revising and commenting on the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The first two authors were supported financially by the National Natural Science Foundation of China (11201473, 11271364) and the Fundamental Research Funds for the Central Universities (2013QNA35, 2010LKSX09, 2010QNA42). The third author was supported financially by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.
References

Cannon, JR: The solution of the heat equation subject to the specification of energy. Q. Appl. Math.. 21, 155–160 (1963)

Ionkin, NI: Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions. Differ. Equ.. 13, 294–304 (1977)

Chegis, RY: Numerical solution of a heat conduction problem with an integral boundary condition. Litov. Mat. Sb.. 24, 209–215 (1984)

Corduneanu, C: Integral Equations and Applications, Cambridge University Press, Cambridge (1991)

Agarwal, RP, O’Regan, D: Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic, Dordrecht (2001)

Yang, ZL: Existence and nonexistence results for positive solutions of an integral boundary value problem. Nonlinear Anal.. 65, 1489–1511 (2006). Publisher Full Text

Yang, ZL: Existence of nontrivial solutions for a nonlinear SturmLiouville problem with integral boundary value conditions. Nonlinear Anal.. 68, 216–225 (2008). Publisher Full Text

Boucherif, A: Secondorder boundary value problems with integral boundary conditions. Nonlinear Anal.. 70, 364–371 (2009). Publisher Full Text

Kong, LJ: Second order singular boundary value problems with integral boundary conditions. Nonlinear Anal.. 72, 2628–2638 (2010). Publisher Full Text

Chamberlain, J, Kong, LJ, Kong, QK: Nodal solutions of boundary value problems with boundary conditions involving RiemannStieltjes integrals. Nonlinear Anal.. 74, 2380–2387 (2011). Publisher Full Text

Feng, MQ: Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. Appl. Math. Lett.. 24, 1419–1427 (2011). Publisher Full Text

Li, YH, Li, FY: Signchanging solutions to secondorder integral boundary value problems. Nonlinear Anal.. 69, 1179–1187 (2008). Publisher Full Text

Li, HT, Liu, YS: On signchanging solutions for a secondorder integral boundary value problem. Comput. Math. Appl.. 62, 651–656 (2011). Publisher Full Text

Jiang, JQ, Liu, LS, Wu, YH: Secondorder nonlinear singular SturmLiouville problems with integral boundary conditions. Appl. Math. Comput.. 215, 1573–1582 (2009). Publisher Full Text

Sun, JX, Zhang, GW: Nontrivial solutions of singular superlinear SturmLiouville problem. J. Math. Anal. Appl.. 313, 518–536 (2006). Publisher Full Text

Han, GD, Wu, Y: Nontrivial solutions of singular twopoint boundary value problems with signchanging nonlinear terms. J. Math. Anal. Appl.. 325, 1327–1338 (2007). Publisher Full Text

Liu, LS, Liu, BM, Wu, YH: Nontrivial solutions of mpoint boundary value problems for singular secondorder differential equations with a signchanging nonlinear term. J. Comput. Appl. Math.. 224, 373–382 (2009). Publisher Full Text

Liu, LS, Liu, BM, Wu, YH: Nontrivial solutions for higherorder mpoint boundary value problem with a signchanging nonlinear term. Appl. Math. Comput.. 217, 3792–3800 (2010). Publisher Full Text

Graef, JR, Kong, LJ: Periodic solutions for functional differential equations with signchanging nonlinearities. Proc. R. Soc. Edinb. A. 140, 597–616 (2010). Publisher Full Text

Wang, YQ, Liu, LS, Wu, YH: Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity. Nonlinear Anal.. 74, 6434–6441 (2011). Publisher Full Text

Krein, MG, Rutman, MA: Linear operators leaving invariant a cone in a Banach space. Transl. Am. Math. Soc.. 10, 199–325 (1962)

Guo, DJ, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Academic Press, Orlando (1988)